Transfer chiles with seeds and soaking liquid to a blender. Add garlic, vinegar, 2 tsp. salt, oregano, cumin, and sugar and purée until a smooth, thick paste forms. Transfer adobo paste to a large bowl; add shrimp and toss until evenly coated.

Preheat broiler. Place shrimp on a rimmed baking sheet in an even layer. Broil, watching closely and stirring halfway through, until shrimp are just cooked through and are browned in spots, 4–5 minutes. Season with salt.